Question

Evaluate:

\[\int\frac{\cos 2x + 2 \sin^2 x}{\sin^2 x}dx\]

Answer 1

\[\int\left( \frac{\cos 2x + 2 \sin^2 x}{\sin^2 x} \right)dx\]
\[ = \int\left( \frac{1 - 2 \sin^2 x + 2 \sin^2 x}{\sin^2 x} \right)dx \left[ \because \cos 2x = 1 - 2 \sin^2 x \right]\]
\[ = \int {cosec}^2\text{ x   dx}\]
\[ = - \cot x + C\]